What is the slope of the line through $(-2,5)$ and $(4,9)$?
Answer (1)
Identify the coordinates of the two given points: ( − 2 , 5 ) and ( 4 , 9 ) .
Apply the slope formula: m = x 2 − x 1 y 2 − y 1 .
Substitute the coordinates into the formula: m = 4 − ( − 2 ) 9 − 5 .
Simplify the expression to find the slope: 3 2 .
Explanation
Understanding the problem and the slope formula. We are given two points through which the line passes: ( − 2 , 5 ) and ( 4 , 9 ) . The slope of a line passing through two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula:
m = x 2 − x 1 y 2 − y 1
where m represents the slope of the line.
Substituting the coordinates into the formula. Let's identify the coordinates:
( x 1 , y 1 ) = ( − 2 , 5 ) and ( x 2 , y 2 ) = ( 4 , 9 ) .
Now, we substitute these values into the slope formula:
m = 4 − ( − 2 ) 9 − 5
Simplifying the expression. Now, let's simplify the expression:
m = 4 − ( − 2 ) 9 − 5 = 4 + 2 4 = 6 4 = 3 2
So, the slope of the line is 3 2 .
Final Answer. Therefore, the slope of the line passing through the points ( − 2 , 5 ) and ( 4 , 9 ) is 3 2 .
Examples
Imagine you're hiking up a hill. The slope represents how steep the hill is. If you know two points on the hill (like your starting and ending points), you can calculate the slope to determine how much your altitude changes for every unit of horizontal distance you travel. A slope of 3 2 means that for every 3 meters you walk horizontally, you climb 2 meters vertically. This concept is crucial in understanding rates of change in various real-world scenarios, from hiking to engineering.
